... | ... | @@ -17,7 +17,7 @@ The different steps of this method are: |
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## 2. Inverse method procedure
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About the first step every detail is given on the dedicated page and nothing has to be added. For the initial guess of the contact force, the $G^2$ value defined in the [gradient analysis](/gradient-analysis) is used. Depending on the quality of the pictures, some manipulation based on this $`G^2`$ can be done in order to distribute the force over the contacts. A good initial force guessing increases the optimization procedure in speed and [results quality](/gradient-analysis#1-overview). The forces determination occurs with an optimization procedure. This procedure is described hereafter as it is specific to the inverse method analysis.
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About the first step every detail is given on the dedicated page and nothing has to be added. For the initial guess of the contact force, the \(G^2\) value defined in the [gradient analysis](/gradient-analysis) is used. Depending on the quality of the pictures, some manipulation based on this \(G^2\) can be done in order to distribute the force over the contacts. A good initial force guessing increases the optimization procedure in speed and [results quality](/gradient-analysis#1-overview). The forces determination occurs with an optimization procedure. This procedure is described hereafter as it is specific to the inverse method analysis.
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### 2.1 Analytical expressions of the photoelastic signal
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... | ... | @@ -36,7 +36,7 @@ The stress field inside the disk is obtained by the superposition of: |
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Of course for the summation, the stress tensors have to be expressed in the same coordinates system.
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In a [circular polariscope](/reflection-photoelasticity#21-photoelasticity-in-the-reflective-polariscope), the light intensity is a function of the difference between the two principal stress. This difference is equal to $`\sigma_{1}-\sigma_{2} = \sqrt{\left(\left(\sigma_{xx}-\sigma_{yy}\right)^2+\left(2\sigma_{xy}\right)^2\right)}`$ which is easely derivate from the [Mohr's circle](https://en.wikipedia.org/wiki/Mohr%27s_circle) for 2D stress.
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In a [circular polariscope](/reflection-photoelasticity#21-photoelasticity-in-the-reflective-polariscope), the light intensity is a function of the difference between the two principal stress. This difference is equal to \(\sigma_{1}-\sigma_{2} = \sqrt{\left(\left(\sigma_{xx}-\sigma_{yy}\right)^2+\left(2\sigma_{xy}\right)^2\right)}\) which is easely derivate from the [Mohr's circle](https://en.wikipedia.org/wiki/Mohr%27s_circle) for 2D stress.
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### 2.2 Guessing initial forces value
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... | ... | @@ -48,7 +48,7 @@ __In the case of high-quality pictures__ (resolution and contrast), a \(G^2_i\) |
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__In the case of low-quality pictures__, each possible combination of active contact is provided to the optimization procedure. The sum of the contact force magnitude is equally distributed over the active contacts. This approach is slower than the previous one and only gives good results when the contact force are almost equally distributed over the active contacts. To overcome the second drawback the fitting procedure will consider all the granular media to propagate the fitted forces on the adjacent disk when the optimization procedure succeeds (PhD Thesis of O. Lantsoght, Université Catholique de Louvain, 2019).
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>JK: I'm currently trying an idea to use betweenness centrality as an initial guess (based on https://pubs.rsc.org/en/content/articlepdf/2018/sm/c8sm01372a ) maybe I can add an outlook here...
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>JK: I'm currently trying an idea to use betweenness centrality as an initial guess (based on [this](https://pubs.rsc.org/en/content/articlepdf/2018/sm/c8sm01372a) ) maybe I can add an outlook here...
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### 2.3 Optimization procedure
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